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title TARGET DECK FILE TAGS tags
Propositional Logic Obsidian::STEM formal-system::propositional
logic
propositional

Overview

Propositional logic is a logical system derived from negation (\neg), conjunction (\land), disjunction (\lor), implication (\Rightarrow), and biconditionals (\Leftrightarrow). A proposition is a sentence that can be assigned a truth value.

%%ANKI Cloze {Propositional} logic is also known as {zeroth}-order logic. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What are the basic propositional logical operators? Back: \neg, \land, \lor, \Rightarrow, and \Leftrightarrow Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is a proposition? Back: A declarative sentence that can be assigned a truth value. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What two categories do propositions fall within? Back: Atomic and molecular propositions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is an atomic proposition? Back: One that cannot be broken up into smaller propositions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is a molecular proposition? Back: One that can be broken up into smaller propositions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Cloze A {molecular} proposition can be broken up into {atomic} propositions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What distinguishes a sentence from a proposition? Back: The latter has an associated truth value. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What are constant propositions? Back: Propositions that contain only constants as operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does Lean define propositional equality? Back: Expressions a and b are propositionally equal iff a = b is true. Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

END%%

%%ANKI Basic How does Lean define propext? Back:

axiom propext {a b : Prop} : (a ↔ b) → (a = b)

Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

END%%

Implication

Implication is denoted as \Rightarrow. In classical logic, it has truth table \begin{array}{c|c|c} p & q & p \Rightarrow q \ \hline T & T & T \ T & F & F \ F & T & T \ F & F & T \end{array}

Implication has a few "equivalent" English expressions that are commonly used. Given propositions P and Q, we have the following equivalences:

  • P if Q
  • P only if Q
  • P is necessary for Q
  • P is sufficient for Q

%%ANKI Basic What name is given to operand a in a \Rightarrow b? Back: The antecedent. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What name is given to operand b in a \Rightarrow b? Back: The consequent. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How do you write "P if Q" in propositional logic? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P if Q" using "necessary"? Back: P is necessary for Q. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P if Q" using "sufficient"? Back: Q is sufficient for P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P only if Q" in propositional logic? Back: P \Rightarrow Q Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P only if Q" using "necessary"? Back: Q is necessary for P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P only if Q" using "sufficient"? Back: P is sufficient for Q. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is necessary for Q" in propositional logic? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is necessary for Q" using "if"? Back: P if Q. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is necessary for Q" using "only if"? Back: Q only if P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is sufficient for Q" in propositional logic? Back: P \Rightarrow Q Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is sufficient for Q" using "if"? Back: Q if P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is sufficient for Q" using "only if"? Back: P only if Q. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P if Q" using "only if"? Back: Q only if P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is sufficient for Q" using "necessary"? Back: Q is necessary for P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P only if Q" using "if"? Back: Q if P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How do you write "P is necessary for Q" using "sufficient"? Back: Q is sufficient for P. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which logical operator maps to "if and only if"? Back: \Leftrightarrow Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Which logical operator maps to "necessary and sufficient"? Back: \Leftrightarrow Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is the converse of P \Rightarrow Q? Back: Q \Rightarrow P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic When is implication equivalent to its converse? Back: It's indeterminate. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is the contrapositive of P \Rightarrow Q? Back: \neg Q \Rightarrow \neg P Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic When is implication equivalent to its contrapositive? Back: They are always equivalent. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given propositions p and q, p \Leftrightarrow q is equivalent to the conjunction of what two expressions? Back: p \Rightarrow q and q \Rightarrow p. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Bibliography